Microwave Susceptibility Observation of Interacting Many-Body Andreev States

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Microwave Susceptibility Observation of Interacting Many-Body Andreev States

V. Fatemi, P. D. Kurilovich, M. Hays, D. Bouman, T. Connolly, S. Diamond, N. E. Frattini, V. D. Kurilovich, P. Krogstrup, J. Nygård, A. Geresdi, L. I. Glazman, and M. H. Devoret

Phys. Rev. Lett. 129, 227701 – Published 21 November 2022

Abstract

Electrostatic charging affects the many-body spectrum of Andreev states, yet its influence on their microwave properties has not been elucidated. We developed a circuit quantum electrodynamics probe that, in addition to transition spectroscopy, measures the microwave susceptibility of different states of a semiconductor nanowire weak link with a single dominant (spin-degenerate) Andreev level. We found that the microwave susceptibility does not exhibit a particle-hole symmetry, which we qualitatively explain as an influence of Coulomb interaction. Moreover, our state-selective measurement reveals a large, $π$ -phase shifted contribution to the response common to all many-body states which can be interpreted as arising from a phase-dependent continuum in the superconducting density of states.

Received 17 December 2021
Accepted 13 September 2022

DOI:https://doi.org/10.1103/PhysRevLett.129.227701

Physics Subject Headings (PhySH)

Andreev reflection Coulomb blockade Fermions Impedance Josephson effect Quasiparticles & collective excitations Superconductivity

Nanowires Quantum dots

Dilution refrigerator Microwave techniques

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

V. Fatemi ^1,*, P. D. Kurilovich¹, M. Hays¹, D. Bouman ^2,3, T. Connolly ¹, S. Diamond¹, N. E. Frattini¹, V. D. Kurilovich¹, P. Krogstrup⁴, J. Nygård ⁵, A. Geresdi ^2,3,6, L. I. Glazman ¹, and M. H. Devoret^1,†

¹Departments of Physics and Applied Physics, Yale University, New Haven, Connecticut 06520, USA
²QuTech and Delft University of Technology, 2600 GA Delft, Netherlands
³Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA Delft, Netherlands
⁴Niels Bohr Institute, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
⁵Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
⁶Quantum Device Physics Laboratory, Department of Microtechnology and Nanoscience, Chalmers University of Technology, SE 41296 Gothenburg, Sweden

^*valla.fatemi@yale.edu
^†michel.devoret@yale.edu

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Vol. 129, Iss. 22 — 23 November 2022

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Images

Figure 1
(a) Schematic of the CQED setup (description in main text), including a colorized scanning electron micrograph of a representative Josephson nanowire weak link (aluminum is dark gray, uncovered region is yellow and 500 nm long in this image, but 350 nm long in the device measured for this report). A probe tone $f_{p}$ reflects with a frequency-dependent reflection amplitude $R$ which is recorded to measure the system. (b) Two-tone spectroscopy as a function of drive frequency and phase bias revealed a single dispersing Andreev transition, with model fit (dashed cyan line, see Fig. 2 for parameters) corresponding to the transition between the even parity states (inset schematic describes the allowed transitions between the many-body eigenstates within the quantum dot model). Parity-switching transitions are forbidden (black arrows with red $X$ ). $n \in {0, 1, 2}$ label the eigenstates. The horizontal lines near 15 and 34 GHz are higher harmonics of the resonator. (c) Schematic of our phenomenological model for the lowest level of the weak link: a single-level quantum dot with charging energy $U$ coupled to superconducting reservoirs with gap $Δ$ and phase drop $φ$ [36].
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Figure 2
(a) Schematic of the resonator frequency shift histogram: while sweeping probe frequency relative to the bare resonator frequency $δ f_{p} = f_{p} - f_{r, b}$ , we record counts when the reflection signal $R$ indicates a resonance. The magnitude of the frequency shift $δ f_{r, n}$ is proportional to the admittance $Y_{n}$ of state $n$ , and normalized counts under each peak give the probability of the state. The widths of the distributions are the same, as they derive from the signal-to-noise ratio of the measurement (see Supplemental Material for details [39]). (b) Resonator frequency shift histogram from the same bias point as Fig. 1. Solid lines are model fits of odd (orange) and even (blue) state dispersive shifts $δ f_{r, n}$ relative to the bare resonator frequency $f_{r, b}$ (white dashed line). The fit parameters in GHz are $Δ / h = 28.6 \pm 0.3$ , $Γ_{L} / h = 6.1 \pm 0.03$ , $Γ_{R} / h = 10.4 \pm 0.06$ , $U / h = 21.6 \pm 0.8$ , and a small offset of $δ f_{off} = (0.045 \pm 0.004) MHz$ to the resonator which is 2% of the resonator linewidth of 2 MHz. The small value for $Δ$ may come from a reduced proximity gap or finite-length effects (see Supplemental Material for discussion [39]). (c) Similar to (b) for bias point 2 in a narrow range of phase (see Fig. S8 for a larger phase range). Black circles mark extracted $δ f_{r, n}$ at $φ = π$ , and the horizontal red bar marks the average of the even-parity frequencies $\frac{1}{2} (δ f_{r, 0} + δ f_{r, 2})$ at $φ = π$ , displaying clear violation of the half-sum rule $\frac{1}{2} (δ f_{r, 0} + δ f_{r, 2}) - δ f_{r, 1} = 0$ (standard deviation is smaller than the red bar thickness). The fit parameters in GHz are $Δ / h = 25 \pm 6$ , $Γ_{L} / h = 7.0 \pm 1.6$ , $Γ_{R} / h = 8.1 \pm 1.8$ , $U / h = 74 \pm 9$ , and $δ f_{off} = (1.8 \pm 1.2) \times 10^{- 4}$ .
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Figure 3
The difference in the occupation probability of the even $n = 0$ and odd $n = 1$ states $P_{even} - P_{odd}$ (i.e., the fermion parity polarization) as a function of $φ$ and $V_{g}$ near gate bias point 2. Panel (a) is the experiment and (b) is a thermal equilibrium prediction within our model for the energies. Both panels have the same pixels and color scale. We assume that the energy of the level in the dot is proportional to $V_{g}$ for simplicity (see Supplemental Material for calibration [39]). (a) Gray regions denote where $δ f_{r, n}$ were too similar to obtain reliable information. Orange and blue stars indicate the extremal experimental polarizations. (b) We use $T = 53 mK$ for the thermal equilibrium prediction (see Supplemental Material for calibration [39]). All other parameters were taken from the fit in Fig. 2, which was taken along the black line.
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